Physics:Quantum Measurement operators - Revision history

WikiHarold: Restore missing Quantum reference definitions

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	A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:		A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:
	<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen"/>		<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen">{{cite book \|last1=Nielsen \|first1=Michael A. \|last2=Chuang \|first2=Isaac L. \|title=Quantum Computation and Quantum Information \|publisher=Cambridge University Press \|year=2000}}</ref>

	After the measurement, the state changes to:		After the measurement, the state changes to:
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	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	== References ==		= References =
	{{reflist\|3}}		{{reflist\|3}}

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	{{Sourceattribution\|Physics:Quantum Measurement operators\|1}}		{{Sourceattribution\|Physics:Quantum Measurement operators\|1}}

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	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
	{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}		{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
	{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}		{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}

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	~~~~{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}Book I

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}

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	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
			{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	'''Measurement operators''' ~~is a Book I topic~~ in ~~the Quantum Collection. In~~ quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. ~~A quantum~~ measurement ~~is described by a set of operators \{M_m\}~~, ~~each associated with a possible outcome m. If~~ the ~~system is in a~~ state ~~\|\psi\rangle, the probability of outcome m is given by the Born rule~~: Measurement operators provide a unified description of different types of quantum measurements: * Projective measurements correspond to projection operators onto eigenstates of an observable.		'''Measurement operators''' in quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. After the measurement, the state changes to: Measurement operators provide a unified description of different types of quantum measurements: Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using Kraus operatorss A_i, such that:
	</div>		</div>

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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	<math display="block">\rho \to \frac{A_i \rho A_i^\dagger}{\operatorname{tr}(\rho E_i)}.</math><ref name="Kraus"/>		<math display="block">\rho \to \frac{A_i \rho A_i^\dagger}{\operatorname{tr}(\rho E_i)}.</math><ref name="Kraus"/>

	Summing over all possible outcomes gives a [[quantum channel]]:		Summing over all possible outcomes gives a quantum channel:
	<math display="block">\rho \to \sum_i A_i \rho A_i^\dagger.</math><ref name="Watrous">{{cite book \|last=Watrous \|first=John \|title=The Theory of Quantum Information \|publisher=Cambridge University Press \|year=2018}}</ref>		<math display="block">\rho \to \sum_i A_i \rho A_i^\dagger.</math><ref name="Watrous">{{cite book \|last=Watrous \|first=John \|title=The Theory of Quantum Information \|publisher=Cambridge University Press \|year=2018}}</ref>

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	<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \\| p_0 \sigma_0 - p_1 \sigma_1 \\|_1.</math><ref name="Helstrom">{{cite book\|first=Carl W.\|last=Helstrom\|title=Quantum Detection and Estimation Theory\|publisher=Academic Press\|year=1976}}</ref>		<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \\| p_0 \sigma_0 - p_1 \sigma_1 \\|_1.</math><ref name="Helstrom">{{cite book\|first=Carl W.\|last=Helstrom\|title=Quantum Detection and Estimation Theory\|publisher=Academic Press\|year=1976}}</ref>

	More generally, optimal measurements can be formulated as optimization problems and solved numerically, for example using [[semidefinite programming]].<ref name="Bae">{{cite journal \|last1=Bae \|first1=Joonwoo \|last2=Kwek \|first2=Leong-Chuan \|title=Quantum state discrimination and its applications \|journal=Journal of Physics A \|volume=48 \|year=2015}}</ref>		More generally, optimal measurements can be formulated as optimization problems and solved numerically, for example using semidefinite programming.<ref name="Bae">{{cite journal \|last1=Bae \|first1=Joonwoo \|last2=Kwek \|first2=Leong-Chuan \|title=Quantum state discrimination and its applications \|journal=Journal of Physics A \|volume=48 \|year=2015}}</ref>

	== Physical interpretation ==		== Physical interpretation ==

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	== Role in the collection ==		== Role in the collection ==
	This page provides a compact reference point for related pages in Book II. It should be read together with nearby matter-scale topics and the corresponding foundations in [[Physics:Quantum mechanics\|quantum mechanics]].<ref name="matter-wiki">{{cite web \|url=https://en.wikipedia.org/wiki/Quantum_mechanics \|title=Quantum mechanics \|website=Wikipedia \|access-date=2026-05-20}}</ref>		This page provides a compact reference point for related pages in Book II. It should be read together with nearby matter-scale topics and the corresponding foundations in [[Physics:Quantum mechanics\|quantum mechanics]].<ref name="matter-wiki">{{cite web \|url=https://en.wikipedia.org/wiki/Quantum_mechanics \|title=Quantum mechanics \|website=Wikipedia \|access-date=2026-05-20}}</ref>

			== Interpretation ==
			For measurement operators, the quantum description is useful because it separates the allowed states, interactions, and measurable quantities from the classical picture. The same concept may appear differently in spectroscopy, scattering, condensed matter, field theory, or cosmology.

			== Related measurements ==
			Typical measurements involve spectra, decay products, transition rates, transport behavior, correlation functions, or detector signatures. These observations provide the empirical link between the page topic and the wider Quantum Collection.

	== See also ==		== See also ==

← Older revision		Revision as of 00:31, 24 May 2026
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	A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:		A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:
	<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen"/>		<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen">{{cite book \|last1=Nielsen \|first1=Michael A. \|last2=Chuang \|first2=Isaac L. \|title=Quantum Computation and Quantum Information \|publisher=Cambridge University Press \|year=2000}}</ref>

	After the measurement, the state changes to:		After the measurement, the state changes to:
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	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	== References ==		= References =
	{{reflist\|3}}		{{reflist\|3}}

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	{{Sourceattribution\|Physics:Quantum Measurement operators\|1}}		{{Sourceattribution\|Physics:Quantum Measurement operators\|1}}

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
	{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}		{{Quantum article nav\|previous=Physics:Quantum Measurement theory\|previous label=Measurement theory\|next=Physics:Quantum Projective measurement\|next label=Projective measurement}}

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	{{Short description\|Quantum Collection topic on Quantum Measurement operators}}		{{Short description\|Quantum Collection topic on Quantum Measurement operators}}

	{{Quantum book backlink\|Measurement and information}}		{{Quantum book backlink\|Measurement and information}}
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~In [[quantum mechanics]],~~ '''~~measurement~~ operators''' provide a general mathematical framework for describing the outcomes of a ~~[[Measurement\|~~measurement]] and the associated change of a [[quantum state]]. They unify different types of quantum measurements, including projective measurements and ~~[[Positive operator-valued measure\|~~POVMs]].~~<ref name="Nielsen">~~{~~{cite book \|last1=Nielsen \|first1=Michael A~~. \|~~last2=Chuang \|first2=Isaac L~~. ~~\|title=Quantum Computation and Quantum Information \|publisher=Cambridge University Press \|year=2000}}</ref>~~		'''Measurement operators''' is a Book I topic in the Quantum Collection. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. In quantum mechanics, measurement operators provide a general mathematical framework for describing the outcomes of a measurement and the associated change of a quantum state. They unify different types of quantum measurements, including projective measurements and POVMs. A quantum measurement is described by a set of operators \{M_m\}, each associated with a possible outcome m. If the system is in a state \|\psi\rangle, the probability of outcome m is given by the Born rule: Measurement operators provide a unified description of different types of quantum measurements: * Projective measurements correspond to projection operators onto eigenstates of an observable.
	</div>		</div>

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	== Introduction ==		== Introduction ==

	A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the [[Born rule]]:		A quantum measurement is described by a set of operators <math>\{M_m\}</math>, each associated with a possible outcome <math>m</math>. If the system is in a state <math>\|\psi\rangle</math>, the probability of outcome <math>m</math> is given by the Born rule:
	<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen"/>		<math display="block">P(m) = \langle \psi \| M_m^\dagger M_m \| \psi \rangle.</math><ref name="Nielsen"/>

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	== State change and Kraus operators ==		== State change and Kraus operators ==

	Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using [[Kraus ~~operator]]s~~ <math>A_i</math>, such that:		Measurement not only yields probabilities but also changes the quantum state. This transformation can be described using Kraus operatorss <math>A_i</math>, such that:
	<math display="block">E_i = A_i^\dagger A_i.</math>		<math display="block">E_i = A_i^\dagger A_i.</math>

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	where <math>p_i</math> is the prior probability of state <math>\sigma_i</math>.		where <math>p_i</math> is the prior probability of state <math>\sigma_i</math>.

	For two states, the optimal measurement is given by the [[Helstrom measurement]]:		For two states, the optimal measurement is given by the Helstrom measurement:
	<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \\| p_0 \sigma_0 - p_1 \sigma_1 \\|_1.</math><ref name="Helstrom">{{cite book\|first=Carl W.\|last=Helstrom\|title=Quantum Detection and Estimation Theory\|publisher=Academic Press\|year=1976}}</ref>		<math display="block">P_{\rm success} = \tfrac12 + \tfrac12 \\| p_0 \sigma_0 - p_1 \sigma_1 \\|_1.</math><ref name="Helstrom">{{cite book\|first=Carl W.\|last=Helstrom\|title=Quantum Detection and Estimation Theory\|publisher=Academic Press\|year=1976}}</ref>